Average Error: 0.5 → 0.4
Time: 12.6s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\frac{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}}
double f(double k, double n) {
        double r161860 = 1.0;
        double r161861 = k;
        double r161862 = sqrt(r161861);
        double r161863 = r161860 / r161862;
        double r161864 = 2.0;
        double r161865 = atan2(1.0, 0.0);
        double r161866 = r161864 * r161865;
        double r161867 = n;
        double r161868 = r161866 * r161867;
        double r161869 = r161860 - r161861;
        double r161870 = r161869 / r161864;
        double r161871 = pow(r161868, r161870);
        double r161872 = r161863 * r161871;
        return r161872;
}
↓
double f(double k, double n) {
        double r161873 = 1.0;
        double r161874 = k;
        double r161875 = sqrt(r161874);
        double r161876 = 2.0;
        double r161877 = atan2(1.0, 0.0);
        double r161878 = r161876 * r161877;
        double r161879 = n;
        double r161880 = r161878 * r161879;
        double r161881 = r161874 / r161876;
        double r161882 = pow(r161880, r161881);
        double r161883 = r161875 * r161882;
        double r161884 = 1.0;
        double r161885 = r161884 / r161876;
        double r161886 = pow(r161880, r161885);
        double r161887 = r161884 * r161886;
        double r161888 = r161883 / r161887;
        double r161889 = r161873 / r161888;
        return r161889;
}

Error

Bits error versus k

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
  2. Using strategy rm

  3. Applied div-sub0.5

    \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}\]

  4. Applied pow-sub0.4

    \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}\]

  5. Applied frac-times0.4

    \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}\]
  6. Using strategy rm

  7. Applied clear-num0.4

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}}}\]

  8. Final simplification0.4

    \[\leadsto \frac{1}{\frac{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))